BACKGROUND OF THE INVENTION
[0001] Turbine engines are driven by a flow of air and combusted gases passing through the
engine onto a multitude of rotating turbine blades.
[0002] A reverse flow combustor can be provided within the engine, the reverse flow combustor
being used to reverse flow twice, once to combust the gas and a second time to exhaust
the gas through the turbine section. Shortening the length of a typical reverse flow
combustor is beneficial in terms of cost and engine efficiency based on weight.
BRIEF DESCRIPTION OF THE INVENTION
[0003] In one aspect, the present disclosure relates to a reverse flow combustion liner
comprising a straight portion having first and second ends, the first end defining
a dome inlet having a predetermined height; a turn portion having third and fourth
ends, the fourth end defining a turbine inlet. The reverse flow combustion line further
includes a dilution portion coupling the second end to the third end and defining
a tangency curve where the dilution portion meets the turn portion, a set of dilution
holes located in the dilution portion upstream of the tangency curve wherein the tangency
curve is a distance less than or equal to 150% of the dome height from the combustor
dome.
[0004] The dilution portion can have an axial length of less than or equal to 20% of the
predetermined height. The dilution portion can have an axial length of less than or
equal to 10% of the predetermined height. The tangency curve can be a distance less
than or equal to 130% of the predetermined height from the dome inlet. The turn portion
can have a decreasing cross-sectional area from the third to the fourth end. The decreasing
cross-sectional area can continuously decrease. A rate at which the continuously decreasing
occurs can be constant. The set of dilution holes can be provided in the combustion
liner axially forward from the tangency curve. The set of dilution holes can be multiple
axially spaced sets of dilution holes circumferentially arranged around the liner.
The combustion liner can comprise an interior liner and an exterior liner defining
a cooling region there between. The set of dilution holes can be provided in the interior
liner and a plurality of cooling holes can be provided on the exterior liner.
[0005] In another aspect, the present disclosure relates to a reverse flow combustor for
a gas turbine engine, the combustor comprising a combustion liner defining a combustor
chamber extending from a dome assembly at a first end and defining a dome height to
a turn defining a tangency curve where an axial length from the first end to the tangency
curve is less than or equal to 150% the dome height.
[0006] A set of dilution holes can be located within a dilution portion located at an axial
distance less than or equal to 110% of the dome height from the first end to define
a second end. The tangency curve can define a third end and a dilution portion extends
from the second end to the third end. The turn can define a turn portion extending
from the third end to a fourth end and the turn portion can have a decreasing cross-sectional
area from the third to the fourth end. The decreasing cross-sectional area can be
continuously decreasing at a constant rate.
[0007] In yet another aspect, the present disclosure relates to a method of mixing dilution
air in a reverse flow combustor, the method comprising: injecting fuel into a combustor
chamber from a dome assembly having a predetermined dome height; igniting the fuel
with an ignitor within the combustor chamber to produce combustion gasses; and injecting
dilution air into the combustor chamber through a set of dilution holes forward of
a tangency curve between a turn portion and a straight portion of the combustor chamber
and wherein the tangency curve is a distance less than or equal to 150% of the dome
height from the combustor dome.
[0008] The method can further include mixing the cooling air with the combustion gasses
in the turn portion of the combustor chamber. The set of dilution holes can be located
within a dilution portion, wherein the dilution portion has an axial length of less
than or equal to 20% of the predetermined height from the tangency curve.
BRIEF DESCRIPTION OF THE DRAWINGS
[0009] In the drawings:
FIG. 1 is a schematic cross-sectional diagram of a free turbine engine for an aircraft
with a reverse flow combustor.
FIG. 2 is an enlarged cross-sectional view of a reverse flow combustor from the prior
art.
FIG. 3 is an enlarged cross-sectional view of a reverse flow combustor for the free
turbine engine of FIG. 1.
FIG. 4 is another cross-sectional view of the reverse flow combustor of FIG. 3 illustrating
varying combustor widths.
DETAILED DESCRIPTION OF THE INVENTION
[0010] Aspects of the disclosure described herein are directed to a reverse flow combustor.
For purposes of illustration, the present disclosure will be described with respect
to a free turbine engine. It will be understood, however, that aspects of the disclosure
described herein are not so limited and that a reverse flow combustor as described
herein can be implemented in other turbine engines, including but not limited to turbojet,
turboprop, turboshaft, and turbofan engines. Aspects of the disclosure discussed herein
may have general applicability within non-aircraft engines having a reverse flow combustor,
such as other mobile applications and non-mobile industrial, commercial, and residential
applications.
[0011] As used herein, the term "forward" or "upstream" refers to moving in a direction
toward the engine inlet, or a component being relatively closer to the engine inlet
as compared to another component. The term "aft" or "downstream" used in conjunction
with "forward" or "upstream" refers to a direction toward the outlet of the engine
or being relatively closer to the engine outlet as compared to another component.
Additionally, as used herein, the terms "radial" or "radially" refer to a dimension
extending between a center longitudinal axis of the engine and an outer engine circumference.
Furthermore, as used herein, the term "set'' or a "set" of elements can be any number
of elements, including only one.
[0012] All directional references (e.g., radial, axial, proximal, distal, upper, lower,
upward, downward, left, right, lateral, front, back, top, bottom, above, below, vertical,
horizontal, clockwise, counterclockwise, upstream, downstream, forward, aft, etc.)
are only used for identification purposes to aid the reader's understanding of the
present disclosure, and do not create limitations, particularly as to the position,
orientation, or use of aspects of the disclosure described herein. Connection references
(e.g., attached, coupled, connected, and joined) are to be construed broadly and can
include intermediate members between a collection of elements and relative movement
between elements unless otherwise indicated. As such, connection references do not
necessarily infer that two elements are directly connected and in fixed relation to
one another. The exemplary drawings are for purposes of illustration only and the
dimensions, positions, order and relative sizes reflected in the drawings attached
hereto can vary.
[0013] Referring to FIG. 1, an engine 10 has a generally longitudinally extending axis or
centerline 12 extending forward 14 to aft 16. The engine 10 is a free-turbine engine
where air is taken in at an inlet 18 and flows in a general aft 16 to forward 14 direction.
The engine 10 includes, in downstream serial flow relationship, a compressor section
22 including the inlet 18, a compressor 24 and a centrifugal compressor 26, a combustion
section 28 including a combustor 30, a turbine section 32 including an HP turbine
34 and an LP turbine 36, an exhaust section 38, and a propeller section 40. The propeller
section 40 includes a propeller hub 42 having a plurality of propeller blades 44 disposed
radially about the centerline 12. An engine casing 46 can form an annular casing for
each of the sections.
[0014] An HP shaft or spool 48 disposed coaxially about the centerline 12 of the engine
10 drivingly connects the HP turbine 34 to the compressor 24 and centrifugal compressor
26. An LP shaft or spool 50, which is disposed coaxially about the centerline 12 of
the engine 10 in line with and separate from the HP spool 48, drivingly connects the
LP turbine 36 to the propeller hub 42. The spools 48, 50 are rotatable about the engine
centerline 12 and couple to a plurality of rotatable elements, which can collectively
define a rotor 51.
[0015] The compressor 24 includes at least one compressor stage 54, in which a set of compressor
blades 56 rotate relative to a corresponding set of static compressor vanes 60 (also
called a nozzle) to compress or pressurize the stream of fluid passing through the
stage. The centrifugal compressor 26 can include an impeller 52 having a set of impeller
blades 58. In a single compressor stage 54 multiple compressor blades 56 can be provided
in a ring and can extend radially outwardly relative to the centerline 12, from a
blade platform to a blade tip, while the corresponding static compressor vanes 60
are positioned upstream of and adjacent to the rotating blades 56. It is noted that
the number of blades, vanes, and compressor stages shown in FIG. 1 were selected for
illustrative purposes only, and that other numbers are possible.
[0016] The blades 56, 58 for a stage of each compressor 24, 26 mount to a disk 61, which
mounts to the corresponding HP spool 48, with each stage having its own disk 61. The
vanes 60 for a stage of the compressor mount to the engine casing 46 in a circumferential
arrangement.
[0017] The HP turbine 34 and the LP turbine 36 respectively include a plurality of turbine
stages 64, 66, in which a set of turbine blades 68, 70 are rotated relative to a corresponding
set of static turbine vanes 72, 74 (also called a nozzle) to extract energy from the
stream of fluid passing through the stage. In a single turbine stage 64, 66, multiple
turbine blades 68, 70 can be provided in a ring and can extend radially outwardly
relative to the centerline 12, from a blade platform to a blade tip, while the corresponding
static turbine vanes 72, 74 are positioned upstream of and adjacent to the rotating
blades 68, 70. It is noted that the number of blades, vanes, and turbine stages shown
in FIG. 1 were selected for illustrative purposes only, and that other numbers are
possible.
[0018] The blades 68 for the HP turbine stages 64 mount to a disk 71, which mounts to the
corresponding HP spool 48, with each stage having a dedicated disk 71. The blades
70 for the LP turbine stages 66 mount to a disk 73, which mounts to the corresponding
LP spool 50, with each stage having a dedicated disk 73. The vanes 72, 74 for a stage
of the respective turbine 34, 36 can mount to the engine casing 46 in a circumferential
arrangement.
[0019] Complementary to the rotor portion, the stationary portions of the engine 10, such
as the static vanes 60, 72, 74 among the compressor and turbine sections 22, 32 are
also referred to individually or collectively as a stator 63. As such, the stator
63 can refer to the combination of non-rotating elements throughout the engine 10.
[0020] In operation, the airflow entering the inlet 18 is channeled into the compressor
24, which then supplies pressurized air 76 to the centrifugal compressor 26, which
further pressurizes the air. The pressurized air 76 from the centrifugal compressor
26 mixes with fuel in the combustor 30 where the fuel combusts, thereby generating
combustion gases. The HP turbine 34 extracts some work from these gases, which drives
the compressor 24 and centrifugal compressor 26. The HP turbine 34 discharges the
combustion gases into the LP turbine 36, which extracts additional work to drive the
propeller hub 42, and the exhaust gas is ultimately discharged from the engine 10
via the exhaust section 38.
[0021] A portion of the pressurized airflow 76 can be drawn from the compressor section
22 as bleed air 77. The bleed air 77 can be drawn from the pressurized airflow 76
and utilized in other areas of the engine or aircraft, such as provided to engine
components requiring cooling. The temperature of pressurized airflow 76 entering the
combustor 30 is significantly increased relative to that of the airflow entering at
the inlet 18. As such, cooling provided by the bleed air 77 is necessary for operating
of such engine components in the heightened temperature environments.
[0022] The LP turbine 36 drives the LP spool 50 to rotate the propeller hub 42 and in turn
the propeller blades 44. A reduction gearbox 80 is provided between the LP spool 50
and the propeller hub 42. Additionally an accessory gear box 82 can be provided to
power other parts of the engine 10, which may include by way of non-limiting example,
fuel pumps, fuel control, oil pumps, a starter/generator, and a tachometer.
[0023] FIG. 2 depicts a cross-section view of a reverse flow combustor 30 according to the
prior art. The reverse flow combustor 30 is disposed radially outwardly of the HP
turbine 34 and axially upstream and forward of the LP turbine 36. The reverse flow
combustor 30 can include an annular liner assembly 100 and a dome assembly 102 defining
a combustor chamber 103, and a plurality of fuel injectors 104 fluidly coupled to
the combustor chamber 103. The reverse flow combustor 30 is disposed within a combustor
casing 106.
[0024] An inner portion 108 of the annular liner assembly 100 can surround the HP turbine
34. An outer portion 110 of the annular liner assembly 100 along with the combustor
casing 106 defines at least a portion of a compressed air passageway 112 through which
compressed air 113 can flow from the compressor section 22 to the reverse flow combustor
30. A plurality of dilution holes 114 can be provided in the annular liner assembly
100 allowing passage of at least a portion of the compressed air 113 from the compressed
air passageway 112 to the combustor chamber 103.
[0025] The plurality of fuel injectors 104 is coupled to and disposed within the dome assembly
102 at a dome inlet 105 comprising a flare cone 107. The dome inlet 105 defines a
dome height (DH). The plurality of fuel injectors 104 includes a fuel inlet 116 that
is adapted to receive a flow of fuel 118, a fuel outlet 120 that is in fluid communication
with the combustor chamber 103, and a linear fuel passageway 122 extending therebetween.
It should be appreciated that one or more of the fuel injectors 104 can have different
characteristics and that the plurality of fuel injectors 104 is for illustrative purposes
only and is not intended to be limiting. A swirler 124 can be provided at the dome
inlet 105 to swirl incoming air in proximity to the fuel exiting the fuel injector
104 and provide a homogeneous mixture of air and fuel entering the combustor.
[0026] The compressed air 113 and fuel 118 is supplied to the combustor chamber 103 via
the dome inlet 105, and is then ignited within the combustor chamber 103 by one or
more igniters 126 to thereby generate combustion gas 128. The combustion gas 128 is
mixed using additional air supplied through the plurality of dilution holes 114 provided
in the inner and outer portions 108, 110 of the annular liner assembly 100 of the
combustor chamber 103, and mixes within a dilution portion 132 of the combustor chamber
103, after which the combustion gas 128 flows through a turn portion 134 and exits
into the HP turbine 34. A first and second radii R1, R2 meet the inner and outer portions
108, 110 at the tangency curve 162 such that the curve is defined as the annular line
at which the dilution portion 132 meets the turn portion 134. The dilution portion
132 can also be called a dilution zone or a mixing zone and have an axial length (DL).
The gas exhausted from the HP turbine 34 is then directed into the LP turbine 36.
A tangency curve 162 defines the beginning of the dilution portion 132. An axial length
(AL) measured from the flare cone 107 to the tangency curve 162 is at least 180% of
the dome height (DH).
[0027] FIG. 3 illustrates a reverse flow combustor 130 according to a preferred aspect of
the disclosure described herein. The reverse flow combustor 130 has similarities to
the reverse flow combustor 30 of the prior art, therefore, like parts will be identified
with like numerals increased by 100. It should be understood that the description
of the like parts of the reverse flow combustor 30 apply to those of the reverse flow
combustor 130, unless otherwise noted.
[0028] An annular liner assembly 200 and a dome assembly 202 define a combustor chamber
203, and a plurality of fuel injectors 204 is fluidly coupled to the combustor chamber
203. The annular liner assembly 200 can include an interior liner 250 and an exterior
liner 252 defining a cooling region 254 there between. A plurality of cooling holes
256 can be provided throughout the exterior liner 252.
[0029] A set of dilution holes 214 is provided in the interior liner 250 and can also be
provided in the exterior liner 252 such that the set of dilution holes 214 circumscribe
and fluidly couple a compressed air passageway 212 with the combustor chamber 203.
The set of dilution holes 214 can be a row or multiple rows of dilution holes 214
axially spaced from each other and annularly disposed around the annular liner assembly
200. In a preferred aspect of the disclosure, the set of dilution holes 214 is a single
row of dilution holes, however, additional smaller or "trim" holes may be applied
in additional rows if required. It should be understood that the annular liner assembly
200 can be a single liner wherein the single annular liner includes the set of dilution
holes 214.
[0030] The annular liner assembly 200 includes a straight portion 260, a dilution portion
232, and a turn portion 234. The straight portion 260 extends from a first end (A)
at the dome assembly 202 defined by a flare cone 207. The straight portion 260 can
also be called a primary zone or igniting zone. The dome assembly 202 defines a dome
inlet 205 having a predetermined height or dome height (H), defining a diameter at
the first end (A). The straight portion 260 terminates just forward of the plane of
the dilution holes 214 in the dilution portion 232 at a second end (B) to define an
axial length (S) equal to 100% of the dome height (H) from the flare cone 207. It
is contemplated that the axial length (S) of the straight portion 260 can between
90% and 130% of the dome height (H).
[0031] The second end (B) is located just forward of the set of dilution holes 214 such
that the dilution holes 214 are within the dilution portion 232. The dilution portion
232 defines an axial length (T) equal to or less than 30% of the dome height (H).
It is further contemplated that the axial length (T) of the dilution portion 232 can
be 20% of the dome height (H). The dilution portion 232 extends from the second end
(B) to a third end (C) to define the axial length (T).
[0032] The third end (C) defines a tangency curve 262 from where the turn portion 234 begins.
A first and second radii R1, R2 meet the inner and outer portions 208, 210 at the
tangency curve 262 such that the curve is defined as the annular line at which the
dilution portion 232 meets the turn portion 234. The tangency curve 262 can be located
at an axial distance (S+T) 130% of the dome height (H) as measured from the flare
cone 207. It is contemplated that the axial length (S + T) can between 110% and 150%
of the dome height (H).
[0033] The turn portion 234 extends from the third end (C) to a fourth end (D) defining
a combustor outlet 264. The combustor outlet 264 transitions to a turbine inlet 266
for an HP turbine 34 (FIG. 1). The turn portion 234 redirects fuel 218 from a primarily
aft direction within the combustor chamber 203 to a primarily forward direction at
the HP turbine 34.
[0034] It should be understood that the axial measurements made are from exemplary locations
and are not limited to being with measured with respect to the flare cone 207 and
tangency curve 262. It is contemplated that axial measurements can be made from anywhere
within the dome assembly 202 and any point defining a beginning of a turn portion
within the combustor chamber 203. The axial length (T) of the dilution portion 232
can be anywhere from 10% to 30% of the dome height (H) as described herein.
[0035] Turning to FIG. 4, a cross-sectional view of the reverse combustor 130 illustrates
the turn portion 234 with a continuously decreasing cross-sectional area (CA) extending
from the third end (C) to the fourth end (D). The rate at which the cross-sectional
area (CA) decreases can be a constant rate; by way of non-limiting example, CA2 is
7% less than CA1 and CA3 is also 7% less than CA2 and so on. The rate need not be
constant, and the differences between consecutive cross-sectional areas located within
10 degrees of each other with respect to a radial plane and can be 10% or less than
the previous cross-sectional area (CA).
[0036] A method of mixing diluting air 268 in the reverse flow combustor 130 as described
herein includes injecting fuel 218 into the combustor chamber 203 from the dome assembly
202. The fuel 218 is ignited with an ignitor 226 within the combustor chamber 203
to produce combustion gasses 228. The method then includes injecting diluting air
268 into the combustor chamber 203 through the set of dilution holes 214 located in
the dilution portion 232 forward of the tangency curve 262 between the turn portion
234 and the straight portion 260 of the combustor chamber 203. The axial length (T)
of the dilution portion 232 is significantly less than the axial length (DL) of the
dilution portion 132 for combustion chamber 103 (FIG. 2). The method therefore further
includes mixing the diluting air 268 with the combustion gasses 228 in the turn portion
234 of the combustor chamber 203.
[0037] Aspects of the disclosure as described herein are directed towards a reverse flow
combustor designed in order to minimize an overall length combusted gases flow within
a combustor chamber. The combustor consists of a conventional straight portion or
primary zone and a short dilution portion or dilution zone. Providing a single row
of openings immediately forward of the tangency curve to the turn portion utilizes
the natural trajectory taken by the diluting air when introduced and maximizes the
amount of mixing between the combusted gases and the diluting air in a small dilution
portion. Additional mixing still takes place in the turn portion and fully mixes prior
to being introduced to the turbine section.
[0038] Conventional reverse flow combustors have a turn portion requiring additional cooled
surface area which requires air that would otherwise be used for dilution and combustion.
This added cooling requirement in a conventional reverse flow combustor limits the
combustor to lower pressure ratios of 12 to 1 or less. Typically dilution and mixing
are complete prior to the gases entering the turning duct, such that the combustor
length ahead of the turn portion is similar to that of a comparable axial combustor.
The reverse flow combustor as described herein enables the final stage of dilution
mixing to occur in the turn portion, therefore allowing for an overall shorter combustor
chamber and consequently reduced cooled surface area requirements.
[0039] Benefits associated with the reverse flow combustor as described herein include reducing
a cooled surface area of the combustor to minimize cooling air required to maintain
liner temperatures within acceptable limits, while still permitting a sufficient primary
zone volume for flame stabilization and ignition. There is also a benefit in terms
of engine weight and packaging by providing an overall shorter combustion liner and
casing. The saving in cooling air requirements permits application of a reverse flow
combustor to a higher pressure ratio cycle, by way of non-limiting example to a 16
to 1 pressure ratio cycle.
[0040] While described with respect to a free turbine propeller engine, it should be appreciated
that the reverse flow combustor as described herein can be for any engine with a high
pressure ratio engine cycle. It should be appreciated that application of aspects
of the disclosure discussed herein are not limited to turbine engines with propeller
sections, but is applicable to engines with fan and booster sections along with turbojets
and turbo engines as well.
[0041] To the extent not already described, the different features and structures of the
various embodiments can be used in combination, or in substitution with each other
as desired. That one feature is not illustrated in all of the embodiments is not meant
to be construed that it cannot be so illustrated, but is done for brevity of description.
Thus, the various features of the different embodiments can be mixed and matched as
desired to form new embodiments, whether or not the new embodiments are expressly
described. All combinations or permutations of features described herein are covered
by this disclosure.
[0042] This written description uses examples to describe aspects of the disclosure described
herein, including the best mode, and also to enable any person skilled in the art
to practice aspects of the disclosure, including making and using any devices or systems
and performing any incorporated methods. The patentable scope of aspects of the disclosure
is defined by the claims, and may include other examples that occur to those skilled
in the art. Such other examples are intended to be within the scope of the claims
if they have structural elements that do not differ from the literal language of the
claims, or if they include equivalent structural elements with insubstantial differences
from the literal languages of the claims.
1. A reverse flow combustion liner comprising:
a straight portion having first and second ends, the first end defining a dome inlet
having a predetermined height;
a turn portion having third and fourth ends, the fourth end defining a turbine inlet;
a dilution portion coupling the second end to the third end and defining a tangency
curve where the dilution portion meets the turn portion;
a set of dilution holes located in the dilution portion upstream of the tangency curve
wherein the tangency curve is a distance less than or equal to 150% of the predetermined
height from the dome inlet.
2. The reverse flow combustion liner of claim 1 wherein the dilution portion has an axial
length of less than or equal to 20% of the predetermined height.
3. The reverse flow combustion liner of claim 1 wherein the dilution portion has an axial
length of less than or equal to 10% of the predetermined height.
4. The reverse flow combustion liner of claim 1 wherein the tangency curve is a distance
less than or equal to 130% of the predetermined height from the dome inlet.
5. The reverse flow combustion liner of claim 1 wherein the turn portion has a decreasing
cross-sectional area from the third to the fourth end.
6. The reverse flow combustion liner of claim 5 wherein the decreasing cross-sectional
area is continuously decreasing.
7. The reverse flow combustion liner of claim 6 wherein a rate at which the continuously
decreasing occurs is constant.
8. The reverse flow combustion liner of claim 1 wherein the set of dilution holes is
provided in the combustion liner axially forward from the tangency curve.
9. The reverse flow combustion liner of claim 1 wherein the set of dilution holes is
multiple axially spaced sets of dilution holes circumferentially arranged around the
liner.
10. The reverse flow combustion liner of claim 1 wherein the combustion liner comprises
an interior liner and an exterior liner defining a cooling region there between.
11. The reverse flow combustion liner of claim 10 wherein the set of dilution holes is
provided in the interior liner and a plurality of cooling holes is provided on the
exterior liner.
12. A reverse flow combustor for a gas turbine engine, the combustor comprising a combustion
liner defining a combustor chamber extending from a dome assembly at a first end and
defining a dome height to a turn defining a tangency curve, where an axial length
from the first end to the tangency curve is less than or equal to 150% of the dome
height.
13. The reverse flow combustor of claim 12 wherein a set of dilution holes is located
within a dilution portion located at an axial distance less than or equal to 110%
of the dome height from the first end to define a second end.
14. The reverse flow combustor of claim 13 wherein the tangency curve defines a third
end and the dilution portion extends from the second end to the third end..
15. The reverse flow combustor of claim 14 wherein the turn defines a turn portion extending
from the third end to a fourth end and the turn portion has a decreasing cross-sectional
area from the third to the fourth end.
16. The reverse flow combustor of claim 15 wherein the decreasing cross-sectional area
is continuously decreasing at a constant rate.
17. A method of mixing cooling air in a reverse flow combustor, the method comprising:
injecting fuel into a combustor chamber from a dome assembly having a predetermined
dome height;
igniting the fuel with an ignitor within the combustor chamber to produce combustion
gasses; and
injecting cooling air into the combustor chamber through a set of dilution holes forward
of a tangency curve between a turn portion and a straight portion of the combustor
chamber and wherein the tangency curve is a distance less than or equal to 150% of
the predetermined dome height from the dome assembly.
18. The method of claim 17 further including mixing the cooling air with the combustion
gasses in the turn portion of the combustor chamber.
19. The method of claim 18 wherein the set of dilution holes are located within a dilution
portion, wherein the dilution portion has an axial length of less than or equal to
20% of the predetermined height from the tangency curve.